Vol 2 - Landau, Lifshitz - The classical theory of fields (4th, 1994) (1) · Problem 10.4

Particle in a Gravitational Field

Physics — Landau & Lifshitz · particle-in-gravitational-field

Problem

A particle of mass

m

moves in a linear potential

V(z) = mgz

for

z > 0

and

V(z) = \infty

for

z < 0

. The energy eigenfunctions are Airy functions. (a) Write down the normalized eigenfunction for the

n

th energy level in terms of the Airy function

\text{Ai}

. (b) Calculate the expectation value

\langle z \rangle

for the

n

th energy level. (c) Calculate

\langle z^2 \rangle

and hence

\Delta z = \sqrt{\langle z^2\rangle - \langle z\rangle^2}